On the Base Point Locus of Surface Parametrizations:Formulas and Consequences

This paper shows that the multiplicity of the base point locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant.As a consequence,we get a new proof of the degree formula relating the degree of the surface,the degree of the parametrization,the base point multiplicity and the degree of the rational map induced by the parametrization.In addition,we extend both formulas to the case of dominant rational maps of the projective plane and describe how the base point loci of a parametrization and its reparametrizations are related.As an application of these results,we explore how the degree of a surface reparametrization is affected by the presence of base points.

IMPROVEMENTS IN HIDDEN LINE REMOVAL ALGORITHM FOR BI-PARAMETRIC SURFACES

A hidden line removal algorithm for bi parametric surfaces is presented and illustrated by some experimental results. The enclosure test is done using area coordinates. A technique of moving box of encirclement is presented. It is found that the algorithm is of general purpose, requires minimal computer storage, has high accuracy and simplicity, and is very easy to be implemented on a computer.

Best bounds on the distance between 3-direction quartic box spline surface and its control net

We present the best bounds on the distance between 3-direction quartic box spline surface patch and its control net by means of analysis and computing for the basis functions of 3-direction quartic box spline surface.Both the local bounds and the global bounds are given by the maximum norm of the first differences or second differences or mixed differences of the control points of the surface patch.

PERIODIC DEFORMATION OF PARAMETRIC SURFACE

Deformation is a powerful tool for geometric modeling and design. Such a toolcan be used to create a new shape from existing shape without restarting whole design process. Anew mathematical model for producing controllable periodic deformations is proposed. By introducingcosine extension functions construct a shape operator matrix and then use the matrix to transformthe position vector of some points on the object surface so as to create the deformation effects.Because the cosine extension functions have a number of variable parameters with differentproperties, the method has corresponding interactive control means. The user can manipulate thoseparameters to get desirable periodic deformation effects. Experimental results show that the methodis feasible and applicable to engineering and research fields such as sheet metal forming bystamping and CAD.

Compact surface plasmon amplifier in nonlinear hybrid waveguide

Surface plasmon polariton（SPP）, a sub-wavelength surface wave promising for photonic integration, always suffers from the large metallic loss that seriously restricts its practical application. Here, we propose a compact SPP amplifier based on a nonlinear hybrid waveguide（a combination of silver, LiNbO3, and SiO2）, where a couple of Bragg gratings are introduced in the waveguide to construct a cavity. This special waveguide is demonstrated to support a highly localized SPP-like hybrid mode and a low loss waveguide-like hybrid mode. To provide a large nonlinear gain, a pumping wave input from the LiNbO3 waveguide is designed to resonate inside the cavity and satisfy the cavity phase matching to fulfill the optical parametric amplification（OPA） of the SPP signal. Proper periods of gratings and the cavity length are chosen to satisfy the impedance matching condition to ensure the high input efficiency of the pump wave from the outside into the cavity. In theoretical calculations, this device demonstrates a high performance in a very compact scheme（~3.32 μm） and a much lower pumping power for OPA compared with single-pass pumping. To obtain a comprehensive insight into this cavity OPA, the influences of the pumping power, cavity length, and the initial phase are discussed in detail.

Shape Modification and Deformation of Parametric Surfaces

A new method for shape modification or deformation of all kinds of parametric surfaces is presented. Instead of using any control mesh and designing any auxiliary tools for modification or deformation, this method only needs to multiply the equation of the original surface by several shape operator matrixes so that deformation effects are achieved. Due to introducing interactive control parameters with different properties those parameters can be changed to get more ideal effects of deformation or shape modification. The core of the method is that so-called cosine extension function is proposed by which the shape operator matrix can be created. Experiment shows that the method is simple, intuitive and easy to control, and diversified local deformations or shape modifications can easily be made via small and random variations of interactive control parameters.

PARAMETRIC SURFACE/SURFACE INTERSECTION

Parametric surfacelsurface intersection methods are essential in a sculptured solid modelingsystem- To improve the robustness, accuracy and topological consistence, an algorithm extendedfrom ideas in [1] and in [3] is described in this paper. Including a new rnethod for obtaining the sur-face near points; an appropriate method for estimating the marching step length; and a reliablernethod for determining singular points. Furthermore, our algorithm can evaluate intersections between offset surfaces without offset approximation. These ideas are discussed and implemented in anintegrated CADICAM system. Tested by rnany typical examples , they have been proved to be robustand efficient. Some exarnples are provided.

RESEARCH AND REALIZATION OF MUTUAL TRIMMING OF NURBS

Since regular surface is defined by a mapping from 2-D parameter plane to 3-D space, trimming of NURBS surface is equivalent to changing a valid parameter region ofthe surface with the same mapping. In this paper, by presenting a rigid and comprehensi-ble definition of union, difference and intersection of two intersecting loops in the parame-ter region of NURBS surface, and by working out the corresponding algorithm, a rebui1d-ing algorithm of the valid parameter region of NURBS surface is obtained.

BOUNDARY PENALTY FINITE ELEMENT METHODS FOR BLENDING SURFACES,I BASIC THEORY

When parametric functions are used to blend 3D surfaces, geometric continuity of displacements and derivatives until to the surface boundary must be satisfied. By the traditional blending techniques, however, arbitrariness of the solutions arises to cause a difficulty in choosing a suitable blending surface. Hence to explore new blending techniques is necessary to construct good surfaces so as to satisfy engineering requirements. In this paper, a blending surface is described as a flexibly elastic plate both in partial differential equations and in their variational equations, thus to lead to a unique solution in a sense of the minimal global surface curvature. Boundary penalty finite element methods (BP-FEMs) with and without approximate integration are proposed to handle the complicated constraints along the blending boundary. Not only have the optimal convergence rate O(h(2)) of second order generalized derivatives of the solutions in the solution domain been obtained, but also the high convergence rate O(h(4)) of the tangent boundary condition of the solutions can be achieved, where h is the maximal boundary length of rectangular elements used. Moreover, useful guidance in computation is discovered to deal with interpolation and approximation in the boundary penalty integrals. A numerical example is also provided to verify perfectly the main theoretical analysis made. This paper yields a framework of mathematical modelling, numerical techniques and error analysis to the general and complicated blending problems.

Experimental investigation on pulse-contrast degradation caused by surface reflection in optical parametric chirped-pulse amplification

In this Letter, we experimentally explore the pulse-contrast degradation caused by surface reflection in optical parameter chirped-pulse amplification. Different pump-to-signal conversion efficiencies and post-pulses with different intensities are obtained by changing the seed-pulse or pump-pulse energy and inserting etalons with different reflection coefficients, respectively. The contrast measurements show that the generated first pre-pulse intensity is proportional to the product of the surface reflection intensity ratio and the square of the pump-to-signal conversion efficiency.

Fast Ray Tracing NURBS Surfaces

In this paper, a new algorithm with extrapolation process for computingthe ray/surface intersection is presented. Also, a ray is defined to be the in-tersection of two planes, which are nonorthogonal in general, in such a waythat the number of multiplication operations is reduced. In the preprocessingstep, NURBS surfaces are subdivided adaptively into rational Bezier patches.Parallelepipeds are used to enclose the respective patches as tightly as possible.Therefore, for each ray that hits the enclosure (i.e., parallelepiped) of a patchthe intersection points with the parallelepiped's faces can be used to vield agood starting poiat for the following iteration. The improved Newton iterationwith extrapolation process saves CPU time by reducing the number of iterationsteps. The intersection scheme is faster than previous methods for which published performance data allow reliable comparison. The method may also beused to speed up tracing the intersection of two parametric surfaces and otheroperations that need Newton iteration.

A study on robotic off-line programming system in induction hardening for fillets or chamfers of mould

As the end-effector of robot, the induction hard- ening tool is required to keep its orientations perpendicular to the fillets or chamfers of the mould at a distance during its uniform motion along the hardening trajectory. This trajectory consists of a group of central curves which are parametric curves on the corresponding chamfers or fillets and between two edges of every chamfer or fillet. The trajectory points and the surface normal vectors at these points are obtained by parametric equations of the fillets or chamfers. This study is conducted to join each central curve into the entire hardening trajectory, including handling on the irregular surfaces and unifying the directions of hardening tool motion. According to kinematics of robot, the trajectory points in modeling coordinate system are transferred into the poses of the induction hardening tool in user frame of robot. The kinematical interference of the induction hardening tool and robot joints is checked by Robognide sim- ulation tool. The validity of the robotic off-line programming （OLP） system was verified by experiments.

Smooth shapes with spherical topology:Beyond traditional modeling,efficient deformation,and interaction

Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces.NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling,continuous surface deformation, shape morphing,reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.

Approximation of the Spectral Fractional Powers of the Laplace-Beltrami Operator

We consider numerical approximation of spectral fractional Laplace-Beltrami problems on closed surfaces.The proposed numerical algorithms rely on their Balakrishnan integral representation and consists a sinc quadrature coupled with standard finite element methods for parametric surfaces.Possibly up to a log term,optimal rate of convergence are observed and derived analytically when the discrepancies between the exact solution and its numerical approximations are measured in L^(2)and H^(1).The performances of the algorithms are illustrated on different settings including the approximation of Gaussian fields on surfaces.